Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.8

The Inverse Football Pool Problem

David Brink
Institut for Matematiske Fag
Københavns Universitet
Universitetsparken 5


The minimal number of spheres (without "interior") of radius n required to cover the finite set {0, ..., q-1}n equipped with the Hamming distance is denoted by T(n,q). The only hitherto known values of T(n,q) are T(n,3) for n = 1, ..., 6. These were all given in the 1950's in the Finnish football pool magazine Veikkaaja along with upper and lower bounds for T(n,3) for n ≥ 7. Recently, Östergård and Riihonen found improved upper bounds for T(n,3) for n = 9,10,11,13 using tabu search. In the present paper, a new method to determine T(n,q) is presented. This method is used to find the next two values of T(n,3) as well as six non-trivial values of T(n,q) with q > 3. It is also shown that, modulo equivalence, there is only one minimal covering of {0,1,2}n for each n ≤ 7, thereby proving a conjecture of Östergård and Riihonen. For reasons discussed in the paper, it is proposed to denote the problem of determining the values of T(n,3) as the inverse football pool problem.

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(Concerned with sequences A004044 A086676.)

Received January 14 2011; revised versions received February 19 2011; June 28 2011; September 5 2011. Published in Journal of Integer Sequences, October 16 2011.

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